Galois lines for the Giulietti–Korchmáros curve

We describe the arrangement of all Galois lines for the Giulietti–Korchmáros curve in the projective 3-space. As an application, we determine the set of all Galois points for a plane model of the GK curve. This curve possesses many Galois points.     

A large class of solutions for the instationary Navier–Stokes system

We investigate very weak solutions to the instationary Navier–Stokes system being contained in where is a bounded domain and . The chosen space of data is small enough to guarantee uniqueness of solutions and existence in case of small data or short time intervals. On the other hand, the data space is large enough that every vector field in is a very weak solution for appropriate data. The solutions and the data depend continuously on each other.      

On the representation of large integers as sums of four almost equal squares of primes

In this paper, we are able to establish two localized results on the conjecture that each large integer congruent to 4 modulo 24 can be written as the sum of four squares of primes. The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method. This work is supported by the National Natural Science Foundation of China (Grant Nos. 10701048 and 10771127).     

Discreteness of the Spectrum of the Laplacian and Stochastic Incompleteness

We propose geometric conditions equivalent to the discreteness of the spectrum of the Laplacian on a class of Riemannian manifolds with ends close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum and stochastic incompleteness.      

Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process

For the Ornstein–Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.     

Order law of large numbers of the Marcinkiewicz–Zygmund type

The order law of large numbers of the Marcinkiewicz–Zygmund type is established for random variables on Banach lattices. Similar results are also obtained for the maximum scheme.     

Wiener–Ikehara theorems and the Beurling generalized primes

The Wiener–Ikehara theorem is classical. Here we prove a converse of the theorem with “if and only if” conditions. We also extend this theorem to cover upper estimates and lower estimates respectively. Significant applications of these theorems to the study of the Beurling generalized primes are introduced too.     

Extension of the Lamé–Gadolin problem for large deformations and its analytical solution

The formulation and solution of the axisymmetric static problem of the stress-strain state of two hollow circular elastic cylinders, one of which is predeformed and inserted into the other cylinder, is considered in the case of large plane deformations (an extension of the Lamé–Gadolin problem to large deformations). Using the theory of the superposition of large deformations, an exact analytical solution of the static problem for cylinders made of incompressible Treloar and Bartenev–Khazanovich materials is obtained, including the case when the cylinders are made of dissimilar materials. An analytical solution is obtained in parametric form for a compressible Blatz–Ko material. Non-linear effects are investigated.     

Global well-posedness for the 3D Navier–Sokes equations with a large component of vorticity

In this paper, we give a new approach to improve the Leray's result concerning the cauchy problem to the 3D Navier–Stokes equations. In particular, global well-posedness with a large component of the initial vorticity is obtained. Our idea is considering the vorticity equations and using some suitable function spaces.     

Global solutions of the Navier–Stokes equations for isentropic flow with large external potential force

We prove the global-in-time existence of weak solutions to the Navier–Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent γ ≥ 1. Initial data and solutions are small in L ² around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when γ = 1. A great deal of information about partial regularity and large-time behavior is obtained.     

A Hardy–Ramanujan-type inequality for shifted primes and sifted sets

Lithuanian Mathematical Journal - We establish an analog of the Hardy–Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we...     

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner-type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.     

Strict Kneser–Poulsen conjecture for large radii

In this paper we prove the Kneser–Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space ${\mathbb{E}^n}$ is rearranged so that the distance between each pair of points does not decrease, then there exists a positive number r ₀ that depends on the rearrangement of the points, such that if we consider n-dimensional balls of radius r > r ₀ with centers at these points, then the volume of the union (intersection) of the balls before the rearrangement is not less (not greater) than the volume of the union (intersection) after the rearrangement. Moreover, the inequality is strict whenever the new point set is not congruent to the original one. Also under the same conditions we prove a similar result about surface volumes instead of volumes. In order to prove the above mentioned results we use ideas from tensegrity theory to strengthen the theorem of Sudakov (Dokl. Akad. Nauk SSSR 197:43–45, 1971), Alexander (Trans. Am. Math. Soc., 288(2):661–678, 1985) and Capoyleas and Pach (Discrete and computational geometry. American Mathematical Society, Providence, 1991), which says that the mean width of the convex hull of a finite number of points does not decrease after an expansive rearrangement of those points. In this paper we show that the mean width increases strictly, unless the expansive rearrangement was a congruence. We also show that if the configuration of centers of the balls is fixed and the volume of the intersection of the balls is considered as a function of the radius r, then the second highest term in the asymptotic expansion of this function is equal to ${-M_nr^{n-1}}$ , where M _n is the mean width of the convex hall of the centers. This theorem was conjectured by Balázs Csikós in 2009.     

Large-time behavior for the solution to the generalized Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space

In this paper, we consider the global existence as well as the optimal decay estimates of the Cauchy problem for the multi-dimensional Benjamin–Bona–Mahony–Burgers equation with large initial data in the whole-space. And these results are obtained by Green?s function method, Fourier analysis method, energy estimates method combined with the time-frequency decomposition method.     

On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.